To summarise:

Robert Tannahill's set his poem to a melody he called "The Three Carls o' Buchannan", dated 1810 and printed soon after, between 1810 and 1814. It was a very popular song to have been printed in song sheet form with music soon after the Poet's death. There was no title for the melody although Tannahill's said his poem was to be sung to The Three Carls o Buchannan. I have a photo copy of Tannahill's letter to R A Smith.

Robert A Smith printed this tune in 1824, three years AFTER he printed The Braes to HIS tune in 1821 which is normally sung today.

Bonaparte was exiled on the Island of St Helena in 1815 so Tannahill's song would be well known along with the tune which was never titled until Smith Indexed as The Three Carles in 1824.

The Scottish words in 'the old soldier's' song suggest it could have been sung to a popular tune which was common then... and still is. It is not surprising therefore, looking at the content, the Bonaparte song melody could be given the title of "Island of St Helena".

Hugh McWilliams published a poem called "The Lass Among the Heather" in 1834 to a melody he suggested was St Helena., 20 years after Tannahill's popular song was written.

Tannahill's poem and McWilliam's were conflated as shown by Jeannie Robertson.
It's entirely possible that because the lyrics were conflated, the two melodies, Smith's and The Three Carles were also conflated hence some printings of "Buonaparte is afar from his war and his fighting.." Social Harp 1855, sound like a mixture of both tunes.

There was a lot of interchange between Paisley and Ireland in 1810 among weavers and Tannahill's was a weaver. Words and tunes were conflated due to the 'folk process' no doubt.

There are many songs with Ballymena or variations of this spelling in Greig/Duncan Collection which use similar lyrics to McWilliam's and Tannahill's but Greig/Duncan never picked up on McWilliam's which I found very strange. Most of the accompanying melodies are variants of the Three Carls tune.

As an aside neither melody has any similarity to McPeaks Wild Mountain Thyme, only the lyrics are taken from Tannahill's but the tune is entirely unique.